Varying slopes? Varying intercepts?

Our dataset has bench and starter performance by team. Let’s look at the relationship between net rating and win percentage by these groups

Ok, varying slopes…big time. What about bench performance only by team?

## `geom_smooth()` using formula = 'y ~ x'

The intercept and slope values are pretty different for benches across different teams. The Nets and Suns have strongly positively correlated bench performance and net ratings, while the Clippers and Warriors have the opposite.

If we simply include a “starter” predictor in our model, we’re estimating different intercepts for starters and bench net ratings–and we know the starter intercept is higher. If we want to know how starter status affects the slope of win percentage, that’s a different specification, one with interactions. The interaction model essentially asks whether or not there is there a difference in the effect of net rating on win percentage for starters and bench players? If we don’t include the interaction we assume that the slope between net rating and win percentage is the same for both starters and bench players.

We could estimate a two-stage regression to estimate the variation by starter status and then by team, or combine these into a multi-level model.

Level	Equation
Level 1	\(winpercentage_{ij} = \beta_{0j} + \beta_{1j}netrating_{ij} + \beta_{2ij}minutes_{ij} + E_{ij}\)
Level 2	\(\beta_{0j} = \gamma_{00} + \gamma_{10}starter_j + U_{0j}\)
	\(\beta_{1j} = \gamma_{10} + \gamma_{11}starter_j + U_{1j}\)
Combined	\(winpercentage_{ij} = \gamma_{00} + \gamma_{01}netrating_{j} + \gamma_{10}minutes_{ij} + \gamma_{20}starter_{ij} + \gamma_{11}minutes_{ij}*starter_j + U_{0j} + U_{1j}starter_{j} + R_{ij}\)

With this model, we will be estimating seven different model parameters — 5 fixed effects, 2 random effects:

\(\gamma_{00}\): the fixed effect for the intercept, controlling for net rating , minutes and starter;
\(\gamma_{01}\): the fixed effect for the slope of net rating, controlling for starter and minutes;
\(\gamma_{10}\): the fixed effect for the slope of minutes, controlling for starter and minutes;
\(\gamma_{20}\): the fixed effect for the slope of starter, controlling for net rating and minutes;
\(\gamma_{11}\): the fixed effect for the cross-level interaction ofminutes with starter, controlling for net rating;
\(\tau_{01}\): a random effect capturing how a team’s mean win percentage varies around the overall mean win percentage, controlling for net rating, minutes and starter;
\(\sigma^2\): a random effect capturing the variance of players around their covariance with team win percentage, controlling for net rating, minutes and starter.

A cross-level interaction is interpreted like an interaction in OLS: the effect of starting status on the effect of net rating on win percentage.

# let's create a new data frame just for our variables of interest and rescale them

df_mod <- df %>% 
  dplyr::select(team_w_pct, min, net_rating, starter_char, season, team_name, player_name, bench) %>% 
  # rescale
  mutate_at(.vars = c(1:3), .funs = arm::rescale) %>%
  arrange(season, player_name)

mlm_1 <- lmer(team_w_pct ~  net_rating + min + bench + min:starter_char + (1|team_name), data = df_mod)

broom.mixed::tidy(mlm_1, conf.int = TRUE)

## # A tibble: 7 × 8
##   effect   group     term        estimate std.error statistic conf.low conf.high
##   <chr>    <chr>     <chr>          <dbl>     <dbl>     <dbl>    <dbl>     <dbl>
## 1 fixed    <NA>      (Intercept)  -0.0361    0.0386    -0.935  -0.112     0.0396
## 2 fixed    <NA>      net_rating    0.358     0.0104    34.3     0.337     0.378 
## 3 fixed    <NA>      min          -0.0663    0.0219    -3.03   -0.109    -0.0234
## 4 fixed    <NA>      bench         0.0454    0.0115     3.94    0.0228    0.0680
## 5 fixed    <NA>      min:starte…   0.107     0.0252     4.26    0.0579    0.156 
## 6 ran_pars team_name sd__(Inter…   0.207    NA         NA      NA        NA     
## 7 ran_pars Residual  sd__Observ…   0.402    NA         NA      NA        NA

We can see from the output above and the plot below that there is a decent amount of uncertainty around our interaction term of interest.

plot_model(mlm_1)

plot_model(mlm_1, type = "int") + theme_classic() +
  labs(x = "minutes Played"
       , y = "Team Win %"
       , title = "Predicted Team Win Percentage by Player Status"
       , caption = "data: nba.com/stats\nwizardspoints.substack.com"
  ) +
  scale_x_continuous(labels = scales::comma_format()) +
  scale_y_continuous(labels = scales::percent_format()) +
  theme(legend.position = "top"
        , legend.title = element_blank()
        , text = element_text(size = 22))

## Scale for y is already present.
## Adding another scale for y, which will replace the existing scale.

Let’s look at the specific team intercepts just for the bench. This will essentially give us a ranking of the interaction.

We can see that for roughly half the league, their benches aren’t doing much for them after we account for variation in net rating by team.

ranef_results <- broom.mixed::tidy(mlm_1, effects = "ran_vals", conf.int = TRUE)

ranef_results %>% 
  arrange(desc(estimate)) %>% 
  ggplot(aes(x = estimate, y = reorder(level, estimate), xmin = conf.low, xmax = conf.high)) + 
  geom_vline(xintercept = 0, linetype = 6) +
  geom_pointrange(aes(col = estimate)) +
  viridis::scale_color_viridis(option = "E") +
  theme_classic() +
  theme(legend.position = "NA") +
  labs(x = "Mean effect of bench minutes", y = "", title = "What is the average effect of bench minutes\nby team on win percentage?", caption = "Data: NBA.com/stats\nwizardspoints.substack.com")

Bring in the original DAG

In our original design, the data are collapsed by starter-bench levels across teams. This still works within a multi-level framework, but likely increases some of the variance of our measures. We should get roughly the same results, but maybe with a little more noise since we are not working at a player-unit level but at a player-status unit level.

df_collapsed <- df %>%
  # drop our character variables
  dplyr::select(team_name, season, bench, net_rating, team_w_pct, min) %>%
  # collapse by team, season, and starter
  group_by(team_name, season, bench) %>% 
  summarize(net_rating = mean(net_rating, na.rm=T)
            , team_w_pct = mean(team_w_pct, na.rm=T)
            , min = sum(min, na.rm=T)
            ) %>% 
  ungroup()  %>% 
  mutate_at(.vars = c("net_rating", "min", "team_w_pct"), .funs = arm::rescale)


mlm_2 <- lmer(team_w_pct~ net_rating + min + bench*min +(1 |team_name), data = df_collapsed)

Our model specification is the same for this model, the data are what have changed.

broom.mixed::tidy(mlm_2, conf.int = TRUE)

## # A tibble: 7 × 8
##   effect   group     term        estimate std.error statistic conf.low conf.high
##   <chr>    <chr>     <chr>          <dbl>     <dbl>     <dbl>    <dbl>     <dbl>
## 1 fixed    <NA>      (Intercept)   -0.138    0.0364     -3.79  -0.209    -0.0666
## 2 fixed    <NA>      net_rating     0.689    0.0268     25.7    0.636     0.741 
## 3 fixed    <NA>      min            0.157    0.0609      2.58   0.0377    0.276 
## 4 fixed    <NA>      bench          0.213    0.0532      4.00   0.108     0.317 
## 5 fixed    <NA>      min:bench     -0.145    0.110      -1.31  -0.361     0.0712
## 6 ran_pars team_name sd__(Inter…    0.105   NA          NA     NA        NA     
## 7 ran_pars Residual  sd__Observ…    0.317   NA          NA     NA        NA

Let’s look at the interaction effect.

plot_model(mlm_2, type = "int") + theme_classic() +
  labs(x = "minutes Played"
       , y = "Team Win %"
       , title = "Predicted Team Win Percentage by Unit Status"
       , caption = "data: nba.com/stats\nwizardspoints.substack.com"
  ) +
  scale_x_continuous(labels = scales::comma_format()) +
  scale_y_continuous(labels = scales::percent_format()) +
  theme(legend.position = "top"
        , legend.title = element_blank()
        , text = element_text(size = 22))

## Scale for y is already present.
## Adding another scale for y, which will replace the existing scale.

## Getting back to our initial DAG model

We want to run a slightly different model on wider data where instead of an interaction term, we use the actual value for bench minutes, controlling for starter and bench net ratings, with random effects by team.

For context, let’s just look at the relationship between bench minutes and win percentage.

df_wide <- df %>% 
  # drop our character variables
  dplyr::select(team_name, season, starter_char, net_rating, team_w_pct) %>% 
  # collapse by team, season, and starter
  group_by(team_name, season, starter_char) %>% 
  summarize(net_rating = mean(net_rating, na.rm=T)
            , team_w_pct = mean(team_w_pct, na.rm=T)
            ) %>%
  ungroup() %>% 
  mutate_at(.vars = c("net_rating",  "team_w_pct"), .funs = arm::rescale) %>% 
  pivot_wider(names_from = starter_char, values_from = net_rating) %>% 
  left_join(
    df %>% 
      # drop our character variables
      dplyr::select(team_name, season, starter, net_rating, min, team_w_pct) %>% 
      group_by(team_name, season, starter) %>% 
      summarize(total_bench_minutes = sum(min, na.rm=T)
                ) %>% 
      filter(starter == 0) %>% 
      dplyr::select(-starter) %>% 
      ungroup() %>% 
      mutate(total_bench_minutes = arm::rescale(total_bench_minutes))
  )

## `summarise()` has grouped output by 'team_name', 'season'. You can override
## using the `.groups` argument.
## `summarise()` has grouped output by 'team_name', 'season'. You can override
## using the `.groups` argument.
## Joining with `by = join_by(team_name, season)`

df %>% 
  # drop our character variables
  dplyr::select(team_name, season, starter_char, net_rating, team_w_pct) %>% 
  # collapse by team, season, and starter
  group_by(team_name, season, starter_char) %>% 
  summarize(net_rating = mean(net_rating, na.rm=T)
            , team_w_pct = mean(team_w_pct, na.rm=T)
            ) %>%
  ungroup() %>% 
  pivot_wider(names_from = starter_char, values_from = net_rating) %>% 
  left_join(
    df %>% 
      # drop our character variables
      dplyr::select(team_name, season, starter, net_rating, min, team_w_pct) %>% 
      group_by(team_name, season, starter) %>% 
      summarize(total_bench_minutes = sum(min, na.rm=T)
                ) %>% 
      filter(starter == 0) %>% 
      dplyr::select(-starter) ) %>% 
      ungroup() %>% 
ggplot(aes(x = total_bench_minutes, y = team_w_pct, col = team_name)) +
  geom_point(shape = 21, stroke = 3, alpha = 0.3) +
  geom_smooth(mapping = aes(group = team_name), method = "lm", se = FALSE, fullrange = TRUE) +
  scale_y_continuous(labels = scales::percent_format()) +
  theme_classic() + 
  theme(legend.position = "NA"
        , legend.title = element_blank()
        ) +
  labs(x = "Total Bench Minutes", y = "Team Win %"
       , title = "Total Bench Minutes and\nOverall Team Win Percentage by Team, 2011-23"
       , caption = "Data: nba.com/stats\nwizardspoints.substack.com"
  ) +
  viridis::scale_color_viridis(discrete = T) +
  facet_wrap(~team_name)

## `summarise()` has grouped output by 'team_name', 'season'. You can override
## using the `.groups` argument.
## `summarise()` has grouped output by 'team_name', 'season'. You can override
## using the `.groups` argument.
## Joining with `by = join_by(team_name, season)`
## `geom_smooth()` using formula = 'y ~ x'

In a model context, this takes the following form: \[ \begin{aligned} \operatorname{team\_w\_pct}_{i} &\sim N \left(\mu, \sigma^2 \right) \\ \mu &=\alpha_{j[i]} + \beta_{1}(\operatorname{Bench}) + \beta_{2}(\operatorname{Starter}) + \beta_{3}(\operatorname{total\_bench\_minutes}) \\ \alpha_{j} &\sim N \left(\mu_{\alpha_{j}}, \sigma^2_{\alpha_{j}} \right) \text{, for team_name j = 1,} \dots \text{,J} \end{aligned} \]

mlm_3 <- lmer(team_w_pct~ Bench + Starter + total_bench_minutes + (1|team_name), data = df_wide)


broom.mixed::tidy(mlm_3, conf.int = TRUE)

## # A tibble: 6 × 8
##   effect   group     term        estimate std.error statistic conf.low conf.high
##   <chr>    <chr>     <chr>          <dbl>     <dbl>     <dbl>    <dbl>     <dbl>
## 1 fixed    <NA>      (Intercept)  -0.0512    0.0123     -4.15 -0.0754    -0.0270
## 2 fixed    <NA>      Bench         0.232     0.0262      8.85  0.181      0.283 
## 3 fixed    <NA>      Starter       0.741     0.0186     39.8   0.705      0.778 
## 4 fixed    <NA>      total_benc…   0.0313    0.0190      1.64 -0.00602    0.0686
## 5 ran_pars team_name sd__(Inter…   0.0397   NA          NA    NA         NA     
## 6 ran_pars Residual  sd__Observ…   0.175    NA          NA    NA         NA

Ok, it looks like our overall mean effect, not split out by team is lower. Let’s look at the intercepts, i.e., the overall team level effects. It’s important to note that we have left the realm of separability; we’re not comparing bench minutes to starter minutes, but estimating the effect of bench minutes after accounting for bench and starter performance, as well as team-level variation.

ranef_results3 <- broom.mixed::tidy(mlm_3, effects = "ran_vals", conf.int = TRUE)

ranef_results3 %>% 
  arrange(desc(estimate)) %>% 
  ggplot(aes(x = estimate, y = reorder(level, estimate), xmin = conf.low, xmax = conf.high)) + 
  geom_vline(xintercept = 0, linetype = 6) +
  geom_pointrange(aes(col = estimate)) +
  viridis::scale_color_viridis(option = "E") +
  theme_classic() +
  theme(legend.position = "NA") +
  labs(x = "Mean effect of bench minutes", y = "", title = "What is the effect of bench minutes by team?", caption = "Data: NBA.com/stats\nwizardspoints.substack.com")

So, we see that for most teams, there is a lot of uncertainty around the mean values (could be positive, could be negative), but with generally the same ranking as the other models, which is kind of reassuring. Notably, there are a couple of teams for whom bench minutes are actively hurting them, on average, but the Wizards aren’t in this group.

Let’s bring in some priors

We know a thing or two about our parameters from watching basketball. Specifically, that starters are good and bench players are typically not as good and bench minutes are not the minutes fans generally go to see (present company excluded).

While Stan defaults to useful, but weakly informative priors, to constrain the uncertainty around the estimates, we can use informative priors based on these assumptions to better estimate the distribution for each parameter. For example, we might think that the bench performance parameter generally has the characteristics \(\beta_{1}(\operatorname{Bench}) \in (-5, 3)\), while starter performance is closer to something like \(\beta_{2}(\operatorname{Starter}) \in (-1, 6)\), and \(\beta_{3}(\operatorname{total bench minutes}) \in (-0.35, -0.01)\) The prior on bench minutes reflects the re-scaled values (i.e., centered and divided by 2 sd’s), so we’re just saying that we thinking bench minutes are mostly negatively correlated with win percentage here.

We can also set a prior on our outcome of win percentage with the following characteristics: \(\sim N \left(0.5, 0.1 \right)\)

If we had thoughts on the distribution of the error of our model, we could also fit a prior to that, but I don’t really have any thoughts on this, so we’ll just keep the default.

prior_mlm <- normal(location = c(-1, 2.5, -0.18), scale = c(4, 3.5, .17), autoscale = FALSE)


prior_intercept_mlm <- normal(location = 0.5, scale = .1, autoscale = F)


mlm_4 <- stan_lmer(team_w_pct~ Bench + Starter + total_bench_minutes + (1|team_name)
                   , data = df_wide, prior = prior_mlm
                   , prior_intercept = prior_intercept_mlm # on second thought lets drop the intercept prior
                   )


broom.mixed::tidy(mlm_4, conf.int = TRUE)

We can see that the median estimate on bench minutes is a bit more constrained (or in normal language, “lower”) than in our previous model. The figure below also uses 90% credible intervals per McElreath.

ranef_results4 <- broom.mixed::tidyMCMC(mlm_4, effects = "ran_vals", conf.int = TRUE, conf.level=.9)

ranef_results4 %>% 
    filter(grepl("team_name:", term)==T & grepl("Sigma", term)!=T) %>% 
    mutate(term = gsub("[_]", " ", substr(term, 25, nchar(term)-1))) %>% 
  arrange(desc(estimate)) %>% 
  ggplot(aes(x = estimate, y = reorder(term, estimate), xmin = conf.low, xmax = conf.high)) + 
  geom_vline(xintercept = 0, linetype = 6) +
  geom_pointrange(aes(col = estimate)) +
  viridis::scale_color_viridis(option = "E") +
  theme_classic() +
  theme(legend.position = "NA") +
  labs(x = "Mean effect of bench minutes", y = "", title = "What is the effect of bench minutes by team?", subtitle = "Estimates using prior information", caption = "Data: NBA.com/stats\nwizardspoints.substack.com")

A linear model

For comparison’s sake, we can see how the multi-level structure compares to a linear model, with the following familiar specification: \[ \operatorname{team\_w\_pct} = \alpha + \beta_{1}(\operatorname{Bench}) + \beta_{2}(\operatorname{Starter}) + \beta_{3}(\operatorname{total\_bench\_minutes}) + \epsilon \]

m1 <- lm(team_w_pct ~ 
           Bench 
         + Starter
         +  total_bench_minutes
         , data = df_wide)

broom::tidy(m1, conf.int = TRUE)

## # A tibble: 4 × 7
##   term                estimate std.error statistic   p.value conf.low conf.high
##   <chr>                  <dbl>     <dbl>     <dbl>     <dbl>    <dbl>     <dbl>
## 1 (Intercept)          -0.0513    0.0102     -5.04 7.45e-  7 -0.0713    -0.0313
## 2 Bench                 0.234     0.0260      9.01 1.31e- 17  0.183      0.285 
## 3 Starter               0.744     0.0184     40.5  2.36e-135  0.708      0.780 
## 4 total_bench_minutes   0.0315    0.0192      1.64 1.02e-  1 -0.00629    0.0692

We can see the multi-level structure at work in the difference overall mean values for bench minutes: lower error, lower magnitude of effect. If we want to re-create the above plot, we can see this difference by team.

df_wide2 <- df_wide %>% bind_cols(predict(m1, interval = "confidence"))

df_wide2 %>% 
  group_by(team_name) %>% 
  summarize(fit = mean(fit, na.rm=T)
            , lwr = mean(lwr, na.rm=T)
            , upr = mean(upr,  na.rm=T)) %>% 
ggplot(aes(x = fit, y = reorder(team_name, fit), xmin = lwr, xmax = upr)) + 
  geom_vline(xintercept = 0, linetype = 6) +
  geom_pointrange(aes(col = fit)) +
  viridis::scale_color_viridis(option = "E") +
  theme_classic() +
  theme(legend.position = "NA") +
  labs(x = "Mean effect of bench minutes", y = "", title = "What is the effect of bench minutes by team?", subtitle = "Linear Regression Results", caption = "Data: NBA.com/stats\nwizardspoints.substack.com")

## A final model \[ \begin{aligned} \widehat{Team Win Percentage}_{i} &\sim N \left(\mu, \sigma^2 \right) \\ \mu &=\alpha_{j[i]} + \beta_{1}(\operatorname{net\_rating}) + \beta_{2}(\operatorname{bench}) + \beta_{3}(\operatorname{min}) + \beta_{4}(\operatorname{bench} \times \operatorname{min}) \\ \alpha_{j} &\sim N \left(\mu_{\alpha_{j}}, \sigma^2_{\alpha_{j}} \right) \text{, for Team j = 1,} \dots \text{,J} \end{aligned} \]

df_wide_pt <- df %>% 
  mutate(bench_char = ifelse(bench == 1, "Bench", "Starter")) %>% 
  # drop our character variables
  dplyr::select(team_name, season, bench_char, bench, net_rating, team_w_pct, min) %>% 
  # collapse by team, season, and starter
  group_by(team_name, season, bench_char, bench) %>% 
  summarize(net_rating = mean(net_rating, na.rm=T)
            , team_w_pct = mean(team_w_pct, na.rm=T)
            , min = sum(min, na.rm=T)
  ) %>%
  ungroup() %>% 
  mutate_at(.vars = c("net_rating",  "team_w_pct", "min"), .funs = arm::rescale)

m_group2 <- df_wide_pt %>% 
  ungroup() %>% 
  nest_by(team_name) %>%
  mutate(fit_win_pct = list(stan_glm(team_w_pct ~ 
                                       net_rating
                                     + bench*min
                                     , data = data))) %>%
  reframe(tidy(fit_win_pct, conf.int = T))

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## SAMPLING FOR MODEL 'continuous' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'continuous' NOW (CHAIN 2).
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## SAMPLING FOR MODEL 'continuous' NOW (CHAIN 3).
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## SAMPLING FOR MODEL 'continuous' NOW (CHAIN 4).
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## SAMPLING FOR MODEL 'continuous' NOW (CHAIN 1).
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## SAMPLING FOR MODEL 'continuous' NOW (CHAIN 2).
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## SAMPLING FOR MODEL 'continuous' NOW (CHAIN 1).
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results_df <- m_group2 %>% 
  filter(term == "bench:min") %>% 
  arrange(desc(estimate)) 

p1 <- results_df %>% 
ggplot(aes(x = estimate, y = reorder(team_name, estimate), xmin = conf.low, xmax = conf.high)) + 
  geom_vline(xintercept = 0, linetype = 6) +
  geom_pointrange(aes(col = estimate), size = 1, linewidth = 1) +
  viridis::scale_color_viridis(option = "viridis") +
  theme_classic() +
  theme(legend.position = "NA"
        , text = element_text(size = 26)) +
  labs(x = "Effect of bench on win percentage (in percentage points)", y = "", title = "What is the effect of bench minutes by team?"
       , caption = "Data: NBA.com/stats\nwizardspoints.substack.com")
p1

# ggsave("02 Output/interaction model results for all teams.png", p1, w = 16, h = 12, dpi = 300)

We can see what the overall effect and the effect for the Wizards:

interact_mod <- glm(team_w_pct ~ 
              net_rating
            +  bench_char*min
            , data = df_wide_pt)

p2 <- interactions::interact_plot(interact_mod, pred = "min", modx = "bench_char", interval = T) +
  labs(x = "Total Minutes", y = "Win Percentage", title = "Relative effect of bench and starter minutes after controlling for performance", caption = "data: nba.com/stats\nwizardspoints.substack.com"
) +
  scale_color_manual(values = c("#E41134", "#00265B")) +
  theme_classic() +
  theme(legend.title = element_blank())

## Scale for colour is already present.
## Adding another scale for colour, which will replace the existing scale.

p2

# ggsave("02 Output/interaction output for all teams.png", p1, w = 12, h = 12, dpi = 300)

Let’s look at the same thing but just for the Wizards.

interact_mod_wiz <- glm(team_w_pct ~ 
              net_rating
            +  bench_char*min
            , data = df_wide_pt[df_wide_pt$team_name == "Washington Wizards",])

p2_wiz <- interactions::interact_plot(interact_mod_wiz, pred = "min", modx = "bench_char", interval = T) +
    labs(x = "Total Minutes", y = "Win Percentage", title = "Relative effect of bench and starter minutes after controlling for performance for the Wizards", caption = "data: nba.com/stats\nwizardspoints.substack.com"
    ) +
    scale_color_manual(values = c("#E41134", "#00265B")) +
    theme_classic() + 
    theme(legend.title = element_blank())

## Scale for colour is already present.
## Adding another scale for colour, which will replace the existing scale.

p2_wiz

# ggsave("02 Output/interaction output for the Wizards.png", p2_wiz, w = 12, h = 12, dpi = 300)

Let’s look at individual effects specifically

# wizards
tidy(stan_glm(team_w_pct ~ 
                net_rating
            +  min
            , data = df_wide_pt[df_wide_pt$team_name== "Washington Wizards" & df_wide_pt$bench==1, ]), conf.int = T)

## 
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## # A tibble: 3 × 5
##   term        estimate std.error conf.low conf.high
##   <chr>          <dbl>     <dbl>    <dbl>     <dbl>
## 1 (Intercept)   0.0866     0.344   -0.543     0.676
## 2 net_rating   -0.420      0.323   -1.01      0.129
## 3 min           0.701      0.672   -0.551     1.81

tidy(stan_glm(team_w_pct ~ 
                net_rating
            +  min
            , data = df_wide_pt[df_wide_pt$team_name== "Golden State Warriors" & df_wide_pt$bench==1, ]), conf.int = T)

## 
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## # A tibble: 3 × 5
##   term        estimate std.error conf.low conf.high
##   <chr>          <dbl>     <dbl>    <dbl>     <dbl>
## 1 (Intercept)   1.42       0.321    0.853     1.99 
## 2 net_rating    0.0180     0.479   -0.817     0.860
## 3 min           2.15       0.692    0.931     3.37

tidy(stan_glm(team_w_pct ~ 
                net_rating
            +  min
            , data = df_wide_pt[df_wide_pt$team_name== "Denver Nuggets" & df_wide_pt$bench==1, ]), conf.int = T)

## 
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## # A tibble: 3 × 5
##   term        estimate std.error conf.low conf.high
##   <chr>          <dbl>     <dbl>    <dbl>     <dbl>
## 1 (Intercept)   -0.443     0.345   -1.01      0.152
## 2 net_rating     0.101     0.242   -0.340     0.504
## 3 min           -1.61      0.800   -2.98     -0.161

tidy(stan_glm(team_w_pct ~ 
                net_rating
            +  min
            , data = df_wide_pt[df_wide_pt$team_name== "Boston Celtics" & df_wide_pt$bench==1, ]), conf.int = T)

## 
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## # A tibble: 3 × 5
##   term        estimate std.error conf.low conf.high
##   <chr>          <dbl>     <dbl>    <dbl>     <dbl>
## 1 (Intercept)   0.174      0.276   -0.310     0.692
## 2 net_rating    0.215      0.458   -0.582     1.04 
## 3 min          -0.0788     0.529   -1.02      0.892

Bench net rating by team

Something that people say online is that good teams have good benches. It’s worth looking at the top net rating seasons for each bench to see what those teams did in that year.

df_wide_pt_no_scale <- df %>% 
  mutate(bench_char = ifelse(bench == 1, "Bench", "Starter")) %>% 
  # drop our character variables
  dplyr::select(team_name, season, bench_char, bench, net_rating, team_w_pct, min) %>% 
  # collapse by team, season, and starter
  group_by(team_name, season, bench_char, bench) %>% 
  summarize(net_rating = mean(net_rating, na.rm=T)
            , team_w_pct = mean(team_w_pct, na.rm=T)
            , min = sum(min, na.rm=T)
  ) %>%
  ungroup() 

df_wide_pt_no_scale %>% 
  filter(bench == 1) %>% 
  group_by(team_name) %>% 
  slice(which.max(net_rating)) %>% 
  arrange(desc(net_rating))

## # A tibble: 30 × 7
## # Groups:   team_name [30]
##    team_name             season  bench_char bench net_rating team_w_pct   min
##    <chr>                 <chr>   <chr>      <dbl>      <dbl>      <dbl> <dbl>
##  1 Chicago Bulls         2011-12 Bench          1      11.5       0.758 5502.
##  2 Toronto Raptors       2019-20 Bench          1       7.36      0.736 5641.
##  3 San Antonio Spurs     2011-12 Bench          1       7.24      0.758 7412.
##  4 Denver Nuggets        2011-12 Bench          1       6.32      0.576 6258.
##  5 Los Angeles Lakers    2020-21 Bench          1       5.78      0.583 7445.
##  6 Indiana Pacers        2020-21 Bench          1       5.12      0.472 6029.
##  7 Memphis Grizzlies     2014-15 Bench          1       4.62      0.671 6191.
##  8 Utah Jazz             2017-18 Bench          1       4.53      0.585 5765.
##  9 Milwaukee Bucks       2011-12 Bench          1       4.36      0.47  6405.
## 10 Oklahoma City Thunder 2022-23 Bench          1       4.34      0.488 6433.
## # ℹ 20 more rows

What we have:

The Bulls bench in 2011-12 had the highest net rating of any bench in their last season making it to the Eastern conference finals.
The Raptors had the best bench the season after they won the championship 19-20 when they made it to the second round of the playoffs
The Spurs best bench in the data was the 11-12 season they made it to the WCF
The Nuggets bench was best in 11-12 when they made it to the first round of the playoffs
the Lakers bench was best in 20-21, when the made it to the first round of the playoffs

###Let’s look at the second and third best seasons, too.

df_wide_pt_no_scale %>% 
  filter(bench == 1) %>%  dplyr::select(season, team_name, net_rating) %>%
  group_by(team_name) %>% 
arrange(desc(net_rating), .by_group = TRUE) %>%
  top_n(3) %>% 
    group_by(team_name) %>% 
    arrange(desc(net_rating), .by_group = TRUE) %>%
    top_n(3) %>% arrange(team_name, desc(net_rating)) %>% mutate(rank = row_number()) %>% arrange(desc(net_rating)) %>% left_join(df_wide_pt_no_scale %>% dplyr::select(season, team_name, team_w_pct)) %>% unique() %>% arrange(desc(team_w_pct)) %>% print(n=30)

## # A tibble: 90 × 5
## # Groups:   team_name [30]
##    season  team_name              net_rating  rank team_w_pct
##    <chr>   <chr>                       <dbl> <int>      <dbl>
##  1 2015-16 Golden State Warriors       0.889     3      0.89 
##  2 2015-16 San Antonio Spurs           6.48      3      0.817
##  3 2017-18 Houston Rockets             1.73      2      0.793
##  4 2021-22 Phoenix Suns                3.46      1      0.78 
##  5 2011-12 Chicago Bulls              11.5       1      0.758
##  6 2011-12 San Antonio Spurs           7.24      1      0.758
##  7 2013-14 San Antonio Spurs           7.11      2      0.756
##  8 2019-20 Toronto Raptors             7.36      1      0.736
##  9 2019-20 Los Angeles Lakers          4.9       2      0.732
## 10 2018-19 Milwaukee Bucks             2.17      3      0.732
## 11 2014-15 Atlanta Hawks               1.73      2      0.732
## 12 2017-18 Toronto Raptors             4.98      2      0.72 
## 13 2013-14 Oklahoma City Thunder       4.29      2      0.72 
## 14 2011-12 Oklahoma City Thunder       1.09      3      0.712
## 15 2020-21 Phoenix Suns                3.03      2      0.708
## 16 2017-18 Golden State Warriors       3.78      1      0.707
## 17 2013-14 LA Clippers                 2.85      3      0.695
## 18 2012-13 Denver Nuggets              1.8       2      0.695
## 19 2015-16 Cleveland Cavaliers         1.19      2      0.695
## 20 2021-22 Memphis Grizzlies           0.664     2      0.683
## 21 2019-20 LA Clippers                 3.35      1      0.681
## 22 2020-21 Philadelphia 76ers         -0.813     2      0.681
## 23 2014-15 Memphis Grizzlies           4.62      1      0.671
## 24 2019-20 Boston Celtics              4.02      2      0.667
## 25 2020-21 Brooklyn Nets               2.75      2      0.667
## 26 2018-19 Denver Nuggets              1.4       3      0.659
## 27 2013-14 Portland Trail Blazers     -0.95      2      0.659
## 28 2020-21 LA Clippers                 3.02      2      0.653
## 29 2018-19 Houston Rockets             4.19      1      0.646
## 30 2021-22 Golden State Warriors       2.31      2      0.646
## # ℹ 60 more rows

Probability of 50 wins given your bench’s net rating is above 0

Let’s do a little Bayes therom \(P(A|B) = \frac{P(B|A) * P(A)}{P(B)}\)

# how many seasons and teams have 50 or more wins?
df_wide_pt_no_scale %>% filter(bench == 1) %>% dplyr::select(team_name, season, team_w_pct) %>% filter(team_w_pct>=0.6097561) %>% count()

## # A tibble: 1 × 1
##       n
##   <int>
## 1    92

# 92 

df_wide_pt_no_scale %>% filter(bench == 1) %>% 
  dplyr::select(team_name, season, net_rating) %>% 
  filter(net_rating>0) %>% count()

## # A tibble: 1 × 1
##       n
##   <int>
## 1    97

# 97 benches have had positive net ratings, 263 had negative net ratings

df_wide_pt_no_scale %>% filter(bench == 1) %>% 
  dplyr::select(team_name, season, net_rating, team_w_pct) %>% 
  filter(net_rating>0 & team_w_pct>=0.6097561) %>% count()

## # A tibble: 1 × 1
##       n
##   <int>
## 1    48

# 48 teams have had a 50+ win season and a bench with a positive net rating

# net rating less than zero and less than 50 wins

df_wide_pt_no_scale %>% filter(bench == 1) %>% 
    dplyr::select(team_name, season, net_rating, team_w_pct) %>% 
    filter(net_rating<0&team_w_pct<0.6097561) %>% count()

## # A tibble: 1 × 1
##       n
##   <int>
## 1   219

# 219

df_wide_pt_no_scale %>% filter(bench == 1) %>% 
    dplyr::select(team_name, season, net_rating, team_w_pct) %>% 
    filter(net_rating<0&team_w_pct>=0.6097561) %>% count()

## # A tibble: 1 × 1
##       n
##   <int>
## 1    44

# 44 teams/seasons have a bench with less than a 0 net rating for their bench, but 50+ wins


p_of_50_wins <- 92/360 # P(B), 360 is the sum of our 97 obs with positive net rating and the 263 with a negative net rating
p_positive = 97/360 # about 13% of teams have had a positive bench net rating, i.e., P(A)

p_positive_bench_and_50_wins = (48/92) # about 6% of teams have had a positive bench net rating and a 50+ win season and we want the conditional , or P(B|A)

p_50wins_given_positive_bench <- (p_positive_bench_and_50_wins*p_of_50_wins)/p_positive
print(p_50wins_given_positive_bench)

## [1] 0.4948454

So there’s about a 49% probability of winning 50 or more games given given that your bench has a positive net rating

df_wide_pt2 <- df_wide_pt_no_scale %>% mutate(fifty = ifelse(team_w_pct>=0.6097561, 1, 0)) %>% filter(bench ==1)

log_m <- glm(fifty~net_rating +min, data = df_wide_pt2, family = "binomial")


exp(coef(log_m)[2])

## net_rating 
##   1.243246

We also see that each one unit increase in bench net rating is associated with a 24% (16-35%) increase in the odds of having a 50+ win season, which is statistically significant

Multi-level model walk-through

Wizards Points

2023-08-08